One-Dimensional Steady-State Conduction
 Conduction problems may involve multiple directions and
time-dependent conditions
 Inherently complex – Difficult to determine temperature
distributions
 One-dimensional steady-state models can represent
accurately numerous engineering systems
 In this chapter we will
 Learn how to obtain temperature profiles for common geometries with
and without heat generation.
 Introduce the concept of thermal resistance and thermal circuits
Chapter 2 : Introduction to Conduction
 For cartesian coordinates
(2.17)
2
Chapter 3 : One-dimensional, Steady state conduction
(without thermal generation)
3.1 Methodology of a conduction analysis
1.
Specify appropriate form of the heat equation
2. Solve for the temperature distribution
3. Apply Fourier’s law to determine the heat flux
Simplest case:
- One-dimensional, steady state conduction with no thermal energy
generation
Common geometries:
i.
The plane wall: described in rectangular (x)
coordinate. Area perpendicular to direction of heat
transfer is constant (independent of x).
ii.
Cylindrical wall : radial conduction through tube
wall
iii.
Spherical wall : radial conduction through shell wall
3
Chapter 3 : One-dimensional, Steady state conduction
(without thermal generation)
3.2 The plane wall – temperature distribution
Using Eq. (2.2) in Chapter 2, by
assuming steady-state conditions and no
internal heat generation (i.e. q = 0), then
the 1-D heat conduction equation reduces
.
to:
For constant k and A, second order
differential equation:
Boundary conditions: T(0) = Ts,1
T(L) = Ts,2
This mean:
Heat flux (q”x) is independent of x
Heat rate (qx) is independent of x
4
Chapter 3 : One-dimensional, Steady state conduction
(without thermal generation)
1-D heat conduction equation for steady-state
. generation (i.e. q
conditions and no internal heat
= 0), is
for constant k and A
Integrate twice to get T(x)
For boundary conditions: T(0) = Ts,1 and T(L) = Ts,2
at x = 0, T(x) = Ts,1 and C2 = Ts,1
at x = L, T(x) = Ts,2 and Ts,2 = C1 L + C2 = C1 L + Ts,1
this gives, C1 = (Ts,2 – Ts,1)/2
Using value of C1 and C2, the function of T(x) is
and
*From here, apply Fourier’s
law to get heat transfer, qx
5
Chapter 3 : One-dimensional, Steady state conduction
(without thermal generation)
Heat flux for plane wall (simplest
case):
Heat rate for plane wall (simplest
case):
6
Chapter 3 : One-dimensional, Steady state conduction
(without thermal generation)
Example: Temp distribution problem
Consider a large plane wall of thickness L = 0.2 m, thermal conductivity k = 1.2
W/mK, and surface area, A = 15m2. The two sides of the wall are maintained at
constant temperatures of T1 = 120C and T2 = 50C. Determine,
a) The temperature distribution equation within the wall
b) Value of temperature at thickness of 0.1m
c) The rate of heat conduction through the wall under steady conditions
7
Thermal Resistance
Based on the previous solution, the conduction heat transfer rate can be
calculated:
T s ,1  T s , 2
dT
kA
(3.2a)
Q x   kA

T s ,1  T s , 2 
dx
L
L / kA
.




Similarly for heat convection, Newton’s law of cooling applies:
.
(T S  T  )
Q x  hA (T S  T  ) 
(3.2b)
1 / hA
And for radiation heat transfer:
.
Q rad  h r A (T s  T sur ) 
(T s  T sur )
(3.2c)
1 / hr A
 Recall electric circuit theory - Ohm’s law for electrical resistance:
Electric current 
Potential Differenc e
Resistance
Thermal Resistance
• We can use this electrical analogy to represent heat transfer problems
using the concept of a thermal circuit (equivalent to an electrical circuit).
.
Q 
Overall Driving Force

Resistance
 T overall
R
 Compare with equations 3.2a-3.2c
 The temperature difference is the “potential” or driving force
for the heat flow and the combinations of thermal
conductivity, convection coefficient, thickness and area of
material act as a resistance to this flow:
R t , cond 
L
kA
, R t , conv 
1
hA
, R t , rad 
1
hr A
Chapter 3 : One-dimensional, Steady state conduction
(without thermal generation)
3.2.1 Thermal resistances & Thermal circuits
- Interestingly, there exists an analogy between the diffusion of heat and electrical
charge. For example if an electrical resistance is associated with the conduction of
electricity, a thermal resistance may be associated with the conduction of heat.
- Defining thermal resistance for conduction in a plane wall:
- For convection :
- For previous simplest case, thermal circuit for plane wall
with adjoining fluids:
10
Chapter 3 : One-dimensional, Steady state conduction
(without thermal generation)
3.2.1 Thermal resistances & Thermal circuits
- In case of radiation :
(3.13)
where,
(1.9)
Surface temperature
Surrounding temperature
11
Chapter 3 : One-dimensional, Steady state conduction
(without thermal generation)
Example: (Problem 3.2a)
The rear window of an automobile is defogged by passing warm air over its inner
surface. If the warm air is at T,i = 40C and the corresponding convection
coefficient is hi = 30 W/m2K, what are the inner and outer surface temperatures of
4-mm thick window glass, if the outside ambient air temperature is T,o = -10C
and the associated convection coefficient is ho = 65 W/m2K.
12
Chapter 3 : One-dimensional, Steady state conduction
(without thermal generation)
Example (problem 3.5):
The walls of a refrigerator are typically constructed by sandwiching a layer of
insulation between sheet metal panels. Consider a wall made from fibreglass
insulation of thermal conductivity, ki = 0.046 W/mK and thickness Li = 50 mm
and steel panels, each of thermal conductivity kp = 60 W/mK and thickness Lp = 3
mm. If the wall separates refrigerated air at T,o = 25C, what is the heat gain per
unit surface area ?
Coefficients associated with natural convection at the inner and outer surfaces can
be approximated as hi = ho = 5 W/m2K
13
Chapter 3 : One-dimensional, Steady state conduction
(without thermal generation)
3.2.2 The composite wall (with negligible contact resistance)
14
Chapter 3 : One-dimensional, Steady state conduction
(without thermal generation)
The composite wall (series type)
Composite wall with negligible contact
resistance:
where,
Overall heat transfer coefficient:
* A modified form of Newton’s Law of cooling
to encompass multiple resistances to heat
transfer
15
Composite Walls
What is the heat transfer rate for this system?
.
Q x  UA  T
Alternatively
R tot 
R
t

T
q

1
UA
where U is the overall heat transfer coefficient and T the overall
temperature difference.
U 
1
R tot A

1
[( 1 / h1 )  ( L A / k A )  ( L B / k B )  ( L C / k C )  (1 / h 4 )]
Chapter 3 : One-dimensional, Steady state conduction
(without thermal generation)
The composite wall (parallel type)
17
Chapter 3 : One-dimensional, Steady state conduction
(without thermal generation)
The composite wall (parallel type)
Electric analogy of thermal
circuits
- To solve a parallel resistance
network like that shown opposite,
we can reduce the network to and
equivalent resistance
For electrical circuits:
For thermal circuits:
18
Chapter 3 : One-dimensional, Steady state conduction
(without thermal generation)
Example: parallel resistances
*IR (infrared) photos show that the heat
transfer through the built-up walls is more
complex than predicted by a simple parallelresistance.
19
Chapter 3 : One-dimensional, Steady state conduction
(without thermal generation)
Example: (3.15)
Consider a composite wall that includes an 8-mm thick hardwood siding, 40 mm
by 130 mm hardwood studs on 0.65 m centers with glass fibre insulation (paper
faced, 28 kg/m3) and a 12 mm layer of gypsum wall board.
What is the thermal resistance associated with a wall that is 2.5 m high by 6.5 m
wide (having 10 studs, each 2.5 m high)
20
Chapter 3 : One-dimensional, Steady state conduction
(without thermal generation)
Example of resistance network with both radiative and convective
boundary (Example 3.1)
21
Contact Resistance
Chapter 3 : One-dimensional, Steady state conduction
(without thermal generation)
3.3 Contact resistance
 It is important to recognise that, in composite systems, the
temperature drop across the interface between material may
be appreciable (present analysis is neglected).
 This attributed is due to thermal contact resistance Rt,c
*values depend on:
materials A and B, surface
finishes, interstitial
conditions and contact
pressure
23
Composite Walls – with contact resistances
Chapter 3 : One-dimensional, Steady state conduction
(without thermal generation)
25
Chapter 3 : One-dimensional, Steady state conduction
(without thermal generation)
26
Chapter 3 : One-dimensional, Steady state conduction
(without thermal generation)
3.3 Radial systems: cylindrical wall
 General heat equation for cylinder (from Chap. 2)
 For 1-D steady state, with no heat generation
 Integrate twice to get temperature
distribution, T(r). For example, for
constant temperature boundary:
 From T(r), heat flux for cylinder
27
Chapter 3 : One-dimensional, Steady state conduction
(without thermal generation)
 The thermal resistance for radial conduction
 In case of cylinder with composite wall (negligible contact resistance)
28
Chapter 3 : One-dimensional, Steady state conduction
(without thermal generation)
Critical radius for insulation

Adding more insulation to a wall  decrease heat transfer

The thicker the insulation, the lower the heat transfer through the wall
 However, adding insulation to a cylindrical pipe or a spherical shell is a
different matter.
 Additional insulation increase the conduction resistance of the insulation
layer but decrease the convection resistance of the surface because of the
increase in the outer surface area for convection
 Hence, knowledge of critical radius of insulation
is required
29
Chapter 3 : One-dimensional, Steady state conduction
(without thermal generation)
Critical radius for insulation: see example 3.5 in Textbook
for details
Insulation prop.
Outside conv. coeff.
 If ri < rcr, Rtot
decreases and the heat
rate therefore
increases with
insulation
 If ri > rcr, Rtot increases
and therefore heat rate
decreases with
insulation
30
Chapter 3 : One-dimensional, Steady state conduction
(without thermal generation)
Example 3.39: cylinder
A stainless steel (AISI 304) tube used to transport a chilled pharmaceutical has an
inner diameter of 36 mm and a wall thickness of 2 mm. The pharmaceutical and
ambient air are at temperatures of 6C and 23C, respectively, while the
corresponding inner and outer convection coefficients are 400 W/m2K and 6
W/m2K, respectively.
i)
What is the heat gain per unit tube length (W/m) ?
ii)
What is the heat gain per unit length if a 10-mm thick layer of calcium silicate
insulation (kins = 0.050 W/mK) is applied to the tube. Discuss the result with
the knowledge of rcrit .
(12.6 W/m, 7.7 W/m)
31
Chapter 3 : One-dimensional, Steady state conduction
(without thermal generation)
3.4 Radial systems: spherical wall
 General heat equation for sphere (from Chap. 2)
 For 1-D steady state, with no heat generation
 Integrate twice to get temperature distribution for
constant k, T(r)
 From T(r), heat flux for sphere
32
Chapter 3 : One-dimensional, Steady state conduction
(without thermal generation)
 The thermal resistance for radial conduction in
sphere
 The total thermal resistance due to conduction and convection in sphere
 In case of sphere with composite shell (negligible contact resistance)
33
Chapter 3 : One-dimensional, Steady state conduction
(without thermal generation)
Summary
34
Chapter 3 : One-dimensional, Steady state conduction
(without thermal generation)
Example 3.54:
A storage tank consists of a cylindrical section that has a length and inner diameter
of L=2m and Di=1m, respectively, and two hemispherical end sections. The tank is
constructed from 20 mm thick glass (Pyrex) and is exposed to ambient air for
which the temperature is 300K and the convection coefficient is 10 W/m2K. The
tank is used to store heated oil, which maintains the inner surface at a temperature
of 400K. Determine the electrical power that must be supplied to a heater
submerged in the oil if the prescribed conditions are to be maintained. Radiation
effects may be neglected, and the Pyrex may be assumed to have a thermal
conductivity of 1.4 W/mK.
35